Hyperintensional logic

Cresswell, M. J.

doi:10.1007/bf02314421

Cited by 129 publications

(58 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…" Carnap [2] generalized Gödel's construction to any formula with one free variable, and proved what today is known as the "diagonalization" or "diagonal lemma". 4 This result can be obtained already in Robinson arithmetic, Q-i.e., PA without induction, plus the axiom ∀x(x = 0 → ∃y(x = Sy)). For all recursive functions are strongly representable in Q. Theorem 1.1 (Diagonalization).…”

mentioning

confidence: 66%

Reference in Arithmetic

Picollo

2018

The Review of Symbolic Logic

View full text Add to dashboard Cite

Abstract. Self-reference has played a prominent role in the development of metamathematics in the past century, starting with Gödel's first incompleteness theorem. Given the nature of this and other results in the area, the informal understanding of self-reference in arithmetic has sufficed so far. Recently, however, it has been argued that for other related issues in metamathematics and philosophical logic a precise notion of self-reference and, more generally, reference, is actually required. These notions have been so far elusive and are surrounded by an aura of scepticism that has kept most philosophers away. In this paper I suggest we shouldn't give up all hope. First, I introduce the reader to these issues. Second, I discuss the conditions a good notion of reference in arithmetic must satisfy. Accordingly, I then introduce adequate notions of reference for the language of first-order arithmetic, which I show to be fruitful for addressing the aforementioned issues in metamathematics. §1. To prove his famous first incompleteness result for arithmetic, Gödel [5] developed a technique called "arithmetization" or "gödelization". It consists in codifying the expressions of the language of arithmetic with numbers, so that the language can 'talk' about its own formulae. Then, he constructed a sentence in the language that he described as stating its own unprovability in a system satisfying certain conditions, 1 and showed this sentence to be undecidable in the system. His method led to enormous progress in metamathematics and computer science, but also in philosophical logic and other areas of philosophy where formal methods became popular. Let's take a closer look.Let L be the language of first-order Peano arithmetic (PA). L contains =, ¬, ∧, ∨, →, ∀, and ∃ as logical constants, 0 as the only individual constant, S as a monadic function symbol, + and × as dyadic function symbols, and a stock of extra function symbols for recursive functions to be specified. All other logical and non-logical symbols are taken to be the usual abbreviations. We assume PA I am deeply indebted to Volker Halbach, with whom I had countless fruitful discussions on reference and self-reference over the last five years. I would also like to particularly thank Thomas Schindler, for great suggestions and encouragement, especially when it came to proofs.

show abstract

mentioning

confidence: 66%

Reference in Arithmetic

Picollo

2018

The Review of Symbolic Logic

View full text Add to dashboard Cite

show abstract

“…e.g. the seminal paper by Cresswell [9] for the notion of hyperintensionality. Meaning is thus an algorithm that computes an expression's denotatum  an idea that has been recently elaborated also by Moschovakis [21] and Muskens [23].…”

Section: Constructions As Hyperintensionsmentioning

confidence: 99%

Existential Import and Relations of Categorical and Modal Categorical Statements

Raclavský

2017

LLP

View full text Add to dashboard Cite

“…We can follow Tichý's rendering of Fregean senses in Transparent Intensional Logic and identify hyperintensions with constructions: each expression will encode the procedure by which its extension is to be computed in each possible world; see (Tichý, 1988;Duží, 2010Duží, , 2012. Other options include identifying hyperintensions with structured intensions (Carnap, 1947;Cresswell, 1975) or algorithms (Moschovakis, 2006). Now, extensions, as the most coarse-grained type of semantic value, are commonly perceived as lying at the bottom of the induced scale.…”

Section: The Semantic Notion Of Formmentioning

confidence: 99%

Extensionality and logicality

Sagi

2017

Synthese

View full text Add to dashboard Cite

Tarski characterized logical notions as invariant under permutations of the domain. The outcome, according to Tarski, is that our logic, which is commonly said to be a logic of extension rather than intension, is not even a logic of extension-it is a logic of cardinality (or, more accurately, of "isomorphism type"). In this paper, I make this idea precise. We look at a scale inspired by Ruth Barcan Marcus of various levels of meaning: extensions, intensions and hyperintensions. On this scale, the lower the level of meaning, the more coarse-grained and less "intensional" it is. I propose to extend this scale to accommodate a level of meaning appropriate for logic. Thus, below the level of extension, we will have a more coarse-grained level of form. I employ a semantic conception of form, adopted from Sher, where forms are features of things "in the world". Each expression in the language embodies a form, and by the definition we give, forms will be invariant under permutations and thus Tarskian logical notions. I then define the logical terms of a language as those terms whose extension can be determined by their form. Logicality will be shown to be a lower level analogue of rigidity. Using Barcan Marcus's principles of explicit and implicit extensionality, we are able to characterize purely logical languages as "sub-extensional", namely, as concerned only with form, and we thus obtain a wider perspective on both logicality and extensionality.

show abstract

Hyperintensional logic

Cited by 129 publications

References 4 publications

Reference in Arithmetic

Reference in Arithmetic

Existential Import and Relations of Categorical and Modal Categorical Statements

Extensionality and logicality

Contact Info

Product

Resources

About